We can think of the unary operations in a lambda-ring as integer linear combinations of Young diagrams; for example the operation $\lambda^n$ corresponds to the Young diagram with $n$ rows and one column.  

Some of these operations are **manifestly non-negative** in the following sense: they're linear combinations of Young diagrams with *natural number* coefficients.  

If we apply a manifestly non-negative operation to an element of the representation ring $R(G)$ coming from a representation of $G$, we get another element coming from a representation - not just a formal difference of such elements.   Similarly, if we apply a manifestly non-negative operation to an element of the K-theory $K(X)$ coming from a vector bundle on $X$, we get another element coming from a vector bundle - not just a formal difference of such elements.

I'm confused about Adams operations.  For $k > 1$, the Adams operation $\psi_k$ is apparently *not* manifestly non-negative, since it's given by an [alternating sum of hook-shaped Young diagrams with $k$ boxes](https://golem.ph.utexas.edu/~distler/blog/archives/001976.html).

And yet, if we apply $\psi_k$ to an element of $R(G)$ coming from a representation $\rho$ of $G$, we get an element coming from a representation, namely the representation $g \mapsto \rho(g^k)$.   

Similarly, if we apply $\psi_k$ to an element of $K(X)$ coming from a vector bundle over $X$, I *believe* we get an element coming from a vector bundle.  It's certainly true for line bundles: $\psi_k [L] = [L^{\otimes k}]$ when $[L]$ is the element of $K$-theory coming from a line bundle $L$.   It's also true for bundles that split as a sum of line bundles, since $\psi_k : K(X) \to K(X)$ is a ring homomorphism.   And I think it follows for all vector bundles using the splitting principle for K-theory ([Corollary 4.3.4 here](https://www.dpmms.cam.ac.uk/~or257/teaching/notes/Kthy.pdf)).

So, it seems that the Adams operations, while not manifestly non-negative, are still **non-negative** in the sense that they send elements of $R(G)$ (resp. $K(X)$) coming from representations (resp. vector bundles) to elements coming from representations (resp. vector bundles) - not merely formal differences of such.

My questions are:

1) Is this true?

2) If so, which integer linear combinations of Young diagrams give operations that are non-negative in this sense?

3) What's really going on here?  In particular, I've defined "non-negative" using $R(G)$ and $K(X)$, but these should be examples of a more general phenomenon.  The Grothendieck ring $K(C)$ of any [symmetric monoidal Cauchy-complete $\mathbb{Q}$-linear category](https://ncatlab.org/johnbaez/show/Schur+functors+I#schur_functors_on_more_general_categories) $C$ is a lambda-ring, in a way that generalizes these two examples.  We can define "non-negative" operations on $K(C)$ to be those sending elements coming from objects of $C$ to elements coming from objects of $C$.   Are Adams operations always non-negative on $K(C)$?  Which linear combinations of Young diagrams give operations that are always non-negative on $K(C)$?